As a special instance of this adversarial model, cGANs [1]} consider where each training sample is associated with some extra information (condition) (e.gclass label [2]}, [3]}, [4]}, text  [5]}, [6]} or an image or a layout [7]}, [8]}, [9]}, [10]}), and the task is to model the conditional distribution \(\mathbb {P}_{\mathbf {x}\mid \mathbf {y}}\)  from a training set of sample pairs \(\lbrace (\mathbf {x}_i,\mathbf {y}_i)\rbrace _{i=1}^N\subseteq {\mathcal {X}}\times {\mathcal {Y}}\)  drawn from the joint distribution \(\mathbb {P}_{\mathbf {x}\mathbf {y}}\)  of observations \(\mathbf {x}\)  and the corresponding conditions \(\mathbf {y}\) . Differently from GANs, in cGANs, the condition information is also fed to both the generator and the discriminator, resulting in the objective:
\(\min _{G} \max _{D}\underset{(\mathbf {x},\mathbf {y})\sim \mathbb {P}_{\mathbf {x}\mathbf {y}}}{\mathbb {E}}[\log D(\mathbf {x},\mathbf {y})]+\underset{\mathbf {z}\sim \mathbb {P}_\mathbf {z},\mathbf {y}\sim \mathbb {P}_\mathbf {y}}{\mathbb {E}}[\log (1-D(G(\mathbf {z},\mathbf {y}),\mathbf {y}))]\) 

